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I Question about the gaps between prime numbers

  1. Jun 25, 2016 #1
    Is there any prime number pn, such that it has a relationship with the next prime number pn+1
    [tex] p_{n+1} > p_{n}^2 [/tex]
    If not, is there any proof saying a prime like this does not exist?

    I have the exact same question about this relation:
    [tex] p_{n+1} > 2p_{n} [/tex]
  2. jcsd
  3. Jun 25, 2016 #2


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    There is also a proof for arbitrary gaps, but see the section "upper bounds".
  4. Jun 25, 2016 #3

    Stephen Tashi

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  5. Jun 26, 2016 #4
    Interesting.Bertrand's Postulate answers the second part of my question. :)

    I see Firoozbakht's conjecture, which is similar to my first part, but it's not quite the same thing as
    [tex] p_{n+1} > p_{n}^2 [/tex]

    I wonder if this can be proved or disproved from other postulates...
  6. Jun 26, 2016 #5
    This also follows very easily from Bertrand's postulate.
  7. Jun 26, 2016 #6

    Stephen Tashi

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    Compare ##2p_n## to ##p^2_n## .
  8. Jun 26, 2016 #7
    Yeah it does. Wow. I'm dumb. :p
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